hensel lifting

This work addresses the problem of improving the efficiency of polynomial factorization over complete discrete valuation rings. By extending Hensel’s lemma within the framework of generalized Newton polygons, the authors propose a novel divide-and-conquer strategy based on approximate root representations to refine the Montes algorithm. Under the assumption that the residual characteristic is zero or sufficiently large, they establish that approximate roots effectively characterize OM-types, thereby achieving nearly optimal factorization complexity. The new method reduces the OM-factorization complexity of a polynomial \( F \) by a factor of \( \delta \), where \( \delta \) denotes the valuation of its discriminant, significantly accelerating both irreducibility testing and factorization compared to the original algorithm.

This work investigates the closure of algebraic complexity classes—VP, VBP, and VNP—under polynomial factorization. Methodologically, it unifies the analysis of Hensel lifting and Newton iteration, precisely delineating their technical equivalence and applicability boundaries, while integrating tools from algebraic circuit complexity and formal power series expansions. The study systematically characterizes structural conditions and fundamental limitations for efficient factorization across these models. Key contributions include the first rigorous demonstration that VP is closed under factorization whereas VNP is not, thereby clarifying VBP’s intermediate status. Several pivotal open problems are posed, notably: “If $f in ext{VNP}$ and a factor $g$ of $f$ lies in VP, must $g$ have polynomially bounded VNP-circuit size?” These results establish a new theoretical framework and benchmark for the intersection of algebraic complexity theory and symbolic computation.

This work addresses the problem of efficiently computing deterministic two-element representations of ideals in number fields. Focusing on ideals whose norm is coprime to the index of the defining polynomial’s ring of integers—a class that includes cryptographically relevant cases such as those defined by cyclotomic polynomials—the paper presents the first deterministic polynomial-time algorithm for this task. The approach leverages a generalized Dedekind criterion to decompose and construct ideals within number fields defined as ℚ[x]/(f). This method overcomes prior limitations that relied on randomization or failed to scale to cryptographic parameters, thereby achieving, for the first time, a combination of determinism, efficiency, and completeness across a broad and practically significant class of ideals used in cryptographic applications.

This work addresses the long-standing challenge of explicitly constructing near-optimal lossless rank extractors, weak subspace designs, and strong $s$-blocking sets over small finite fields whose size depends only on the rank or codimension. By integrating tools from function field theory, polynomial identity testing, and Fourier analysis based on $\varepsilon$-biased sets, the authors achieve the first explicit near-optimal constructions of these objects over non-prime fields with $q \geq \mathrm{poly}(s)$. Notably, the resulting strong $s$-blocking set has size $O(s(k - s)q^s)$, improving upon the previous exponential bound $2^{O(s^2 \log s)} q^s k$ and matching the non-explicit optimal asymptotics. The paper also presents the first explicit near-optimal constructions for both lossless rank extractors and weak subspace designs in this setting.

This study investigates the arithmetic properties of hypergeometric functions over the p-adic numbers, with a focus on their p-adic valuations and reduction behavior modulo primes. Building upon Christol’s theorem and integrating p-adic analysis with algebraic algorithms, the work achieves the first exact computation of p-adic valuations within arbitrary disks of convergence and establishes a systematic, effective criterion for determining the mod-p reducibility of hypergeometric functions. Furthermore, it introduces an algorithm to construct annihilating polynomials for the reductions modulo p. These contributions provide practical computational tools for the theory of arithmetic D-modules and significantly advance the algorithmic understanding of the arithmetic properties of hypergeometric functions.

This work addresses the efficient algebraic representation and factorization of linear ordinary differential operators over compatible derivation modules. By implementing differential operators as first-class objects in Scratchpad II, the approach supports standard notation and provides a unified treatment of left and right module structures. For operators with coefficients in a field or polynomial ring, it integrates Ore localization, pseudo-division, and construction of right fraction fields to enable left and right division, computation of greatest common divisors, least common multiples, and extended Euclidean algorithms. Furthermore, by combining Riccati equations with Newton polygon analysis, the method effectively characterizes the singularities of factors. This framework facilitates constructive factorization and algebraic manipulation of operators with constant, elementary, rational, and even matrix-valued coefficients.

This work addresses the efficient computation of sum-of-squares multipliers (i.e., certificates) for non-negative univariate polynomials within Archimedean saturated quadratic modules, thereby verifying their membership. To this end, the authors propose a novel symbolic algorithm that leverages the natural generators introduced by Kuhlmann and Marshall, incorporates the Basic Lemma to decompose non-negative factors, and employs a systematic case analysis to achieve, for the first time, a constructive transformation from natural to primitive generators. This approach establishes a complete framework for certificate construction in univariate Archimedean saturated quadratic modules. Implementation in Maple demonstrates the algorithm’s effectiveness and superiority, successfully handling several instances where RealCertify fails.

We introduce $p$-adic manifold learning, propose an algorithm to solve it, and propose benchmark tasks from impartial games.

This study investigates the computational complexity of $p$-adic optimization problems beyond the binary ($p=2$) case. By generalizing Baker’s forcing method to arbitrary primes $p$, the authors establish a reduction framework linking integer optimization to $p$-adic optimization. Leveraging this framework, they provide the first unified proof that several prominent $p$-adic models—including $p$-adic linear regression, 2-adic dynamic neural networks, and various van der Put–based $p$-adic neural network architectures—are all NP-hard. This work bridges $p$-adic analysis, computational complexity theory, and neural network modeling, significantly extending the applicability of forcing techniques and offering a foundational complexity-theoretic characterization for $p$-adic machine learning.